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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the punctual and local ergodic theorem for nonpositive power bounded operators in $ L\sp p\sb {{\bf C}}[0,1],\;1<p<+\infty$

Author: I. Assani
Journal: Proc. Amer. Math. Soc. 96 (1986), 306-310
MSC: Primary 47A35; Secondary 47D05
MathSciNet review: 818463
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Abstract: We show in this note that there exists a function $ f \in { \cap _1}_{ < p < + \infty }L_{\mathbf{C}}^p[0,1]$ and for each $ p$ an isomorphism $ T:L_{\mathbf{C}}^p \to L_{\mathbf{C}}^p$ such that $ {\text{su}}{{\text{p}}_{n \in {\mathbf{Z}}}}\left\Vert {{T^n}} \right\Vert < + \infty $ and $ T$ does not satisfy the punctual ergodic theorem.

We give also an example of a one-parameter semigroup $ ({T_t},t \geqslant 0)$ of power bounded operators in each $ L_{\mathbf{C}}^p(1 < p < + \infty )$ for which the assertion of the local ergodic theorem $ ((1/t)\smallint _0^t{T_s}fds$ converge almost everywhere as $ t \to {0_ + }$ for all $ f \in {L^p}$ fails to be true.

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Article copyright: © Copyright 1986 American Mathematical Society